Problem: Nick is doing some math exercises on a website called Khon Academy. In Khon Academy, you have to get at least $70\%$ of the problems in an exercise right in order to gain proficiency. So far, Nick answered one question, and he didn't get it right. Suppose he answers $75\%$ of the following $q$ questions correctly and gains proficiency in the exercise. Write an inequality in terms of $q$ that models the situation.
Answer: The strategy We know that in the end, Nick answered at least $70\%$ of the total number of questions correctly. If we let $C$ denote the number of questions that Nick answered correctly and $T$ denote the total number of questions, we obtain the inequality $\dfrac{C}{T}\geq0.7$. Now, let's express $C$ and $T$ in terms of $q$. Expressing the total number of questions Nick answered his first question incorrectly. He then proceeded to answer $q$ more questions. Therefore, the total number of questions that Nick answered is $q+1$. Expressing the number of correctly answered questions We know that Nick got his first question wrong, but then answered $75\%$ of the next $q$ questions correctly. Therefore he answered $0.75q$ questions correctly. Putting things together We found that $C=0.75q$ and $T=q+1$. Since $\dfrac{C}{T}\geq0.7$, we can substitute and find an inequality in terms of $q$ that models the situation. The answer is: $ \dfrac{0.75q}{q+1}\geq 0.7 $